A new elastoviscoplastic model based on the Herschel-Bulkley viscoplastic model

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A new elastoviscoplastic model based on the Herschel-Bulkley viscoplastic model

Pierre Saramito

To cite this version:

Pierre Saramito. A new elastoviscoplastic model based on the Herschel-Bulkley viscoplastic model. Journal of Non-Newtonian Fluid Mechanics, Elsevier, 2009, 158 (1-3), pp.154-161.

�10.1016/j.jnnfm.2008.12.001�. �hal-00316158v3�

A new elastoviscoplastic model based on the Herschel-Bulkley viscoplastic model

Pierre Saramito December 1, 2008

Abstract – The aim of this paper is to introduce a new three-dimensional elastoviscoplastic model that combines both the Oldroyd viscoelastic model and the Herschel-Bulkley viscoplastic model with a power-law index n >0. The present model is derived to satisfy the second law of thermodynamics. Various fluids of practical interest, such as liquid foams, droplet emulsions or blood, present such elastoviscoplastic behavior: at low stress, the material behaves as a vis- coelastic solid, whereas at stresses above a yield stress, the material behaves as a fluid. When n= 1, a recently introduced elastoviscoplastic model proposed by the author is obtained. When 0< n <1, then the plasticity criteria becomes smooth, the elongational viscosity is always well defined and the shear viscosity shows a shear thinning behavior. This is a major improvement to the previous elastoviscoplastic model. Finally, whenn >1, the material exhibits the unusual shear thickening behavior.

Keywords – non-Newtonian fluid; viscoelasticity; viscoplasticity; constitutive equation.

1 Introduction

In 1926, Herschel and Bulkley [11] proposed a power law variant of the viscoplastic Bingham model [2] :







|τ| ≤τ0 when ˙ε= 0 τ =k|ε˙|ⁿ⁻¹ε˙+τ0

˙ ε

|ε˙| otherwise ⇐⇒max

0, |τ| −τ0

k|τ|ⁿ n¹

τ = ˙ε

where τ is the stress, ˙ε the rate of deformation, k > 0 the consistency parameter and τ0 > 0 the yield stress. Note that k|ε˙|ⁿ⁻¹ has the dimension of a viscosity. Here, n > 0 is the power index. When n= 1 the model reduces to the Bingham model. The shear thinning behavior is associated with 0< n < 1 and the unusual shear thickening behavior ton > 1. In [26], by introducting viscoelasticity into a viscoplastic model, a three-dimensional combination of the viscoelastic Oldroyd and viscoplastic Bingham model has been derived, and has been studied in the context of liquid foam in [5]. The aim of this paper is to explore the elastoviscoplastic extension of the Herschel and Bulkley model, that is written in one-dimensional form:

1

µτ˙+ max

0, |τ| −τ0

k|τ|^{n 1}n

τ = ˙ε (1)

where µ >0 is the elasticity parameter andσ=τ+ηε˙is the total stress, where η >0 is a viscosity, often called the solvent viscosity in the context of polymer melts. When n = 1 and τ0 = 0 we obtain a one- dimensional version of the Oldroyd viscoelastic model [19]. The new model is motivated by the existence of

materials that exhibit viscoelastic solid-like behavior at low stress and power-index shear thinning fluid-like behavior at high stress. Various authors have fitted their rheological data to the Herschel-Bulkley model with 0< n <1 e.g. [12, 14, 15, 16, 22, 29], in a variety of elastoviscoplastic applications that include concentrated Carbopol micro-gel dispersions, liquid foams and biological flows containing cells. Notice also that Laun (see [17, p. 265] and related references) points out the unusual shear thickening behavior (n > 1 case) for suspensions of solid particles.

0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

1111 1111 1111 1111 1111 1111 1111 1111 1111

1111 η

εe

τ0

µ k, n

σ

εp

ε

(a)

¯ σ > τ0

¯ σ≤τ0

n= 0.5 n= 1 n= 1.5 n= 2

(b)

t

0

ε(t)

¯ σ µ

Figure 1: Creeping test for the present model.

The mechanical model is represented in Fig. 1a. At stresses below the yield stress, the friction element remains rigid. The level of the elastic strain energy required to break the friction element is determined by the von Mises yielding criterion. Consequently, before yielding, the whole system predicts only recoverable Kelvin-Voigt viscoelastic deformation due to the springµ and the viscous elementη. The elastic behavior τ =µεis expressed in (1) in differential form, where τ denotes the elastic stress. Before yielding, the total stress isσ=µε+ηε. As soon as the strain energy exceeds the level required by the von Mises criterion, the˙ elastic stress in the friction element attains the yield value and the element breaks allowing deformation of all the other elements. After yielding, the deformation of these elements describes a non-linear viscoelastic behavior.

The evolution in time of elongationε(t) for a fixed imposed tractionσ(creeping) is represented on Fig 1b.

When σ ≤ τ0, the elongation for a fixed imposed traction is bounded in time: the material behaves as a Kelvin-Voigt viscoelastic solid. Otherwise, σ > τ0, the elongation is unbounded in time: the material behaves as a viscoelastic fluid. A number of other closely related models have appeared in the literature and an historical presentation in 2005 is available in [26]. More recently, in 2007, Fusi and Farina [7]

proposed a new model that combines an elastic solid before yielding and a Newtonian fluid after yielding.

Based on an explicit description of the free surface separating the solid and the fluid parts, this model leads to complex formulations and computations, even for some simple shear flows such as the Poiseuille flow.

In 2008, Benito et al. [1] performed a review of the subject and proposed a classification of elastoviscoplastic models. Combining a solid Burger material before yielding and a viscoelastic fluid after yielding, these authors proposed also a variant of the model [26].

The aim of the present article is to build the proposed model for the general three-dimensional case (section 2) and to study it with simple shear and extensional flows (section 3). The impatient reader – and the reader who is unfamiliar with the thermodynamic framework – could jump directly to the end of section 2 where

the complete set of equations (7) governing such a flow is presented, before reading section 3 devoted to applications.

2 The proposed model

2.1 Thermodynamic framework

The state of the system is described by using two independent variables : the total deformation tensor ε and an internal variable, the elastic deformation tensor εe. We have ε=εe+εp where εp represents the plastic deformation tensor. Following Halphen and Nguyen [10] (see e.g. [28] or [18, p. 97]) we say that a generalized standard materialis characterized by the existence of a free energy functionE and a potential of dissipationD, that are both convex functions of their arguments. The proposed model can be written as:

E(ε, εe) = µ|εe|²,

D( ˙ε,ε˙e) = ϕ( ˙ε) +ϕp( ˙ε−ε˙e), (2) whereµ >0 is the elasticity parameter and where|.|denotes the matrix norm, defined by a double contraction of indices : |εe|²=εe:εe. The functionsϕandϕp are expressed by :

ϕ( ˙ε) =

η|ε˙|² when tr ˙ε= 0,

+∞ otherwise, and ϕp( ˙εp) =





 2k

n+ 1|ε˙p|ⁿ⁺¹+τ0|ε˙p| when tr ˙εp= 0,

+∞ otherwise. (3)

When n = 1, the model coincides with the elastoviscoplastic model introduced in [26] that combines the Bingham and the Oldroyd models. The ϕ function expresses the incompressible viscous behavior and is associated with the viscosity η ≥ 0 while the ϕp function expresses the viscoplastic behavior by using a power law index n > 0 and a consistency parameter k > 0, acting on continuous modification of the network links, and also a yield stress valueτ0≥0. When the elastic stress becomes higher than this value, some topological modifications appear in the network of contacts. This model satisfies the second law of thermodynamics: in the framework of generalized standard materials, see [10, 28, 18]. This property is a direct consequence of the convexity of bothE and D.

2.2 The general constitutive law

Let Ω be a bounded domain of R^N, where N = 1,2,3. Since both ϕ and ϕp are non-linear and non- differentiable, the following manipulations involve subdifferential calculus from convex analysis. The material constitutive laws can be written as:

σ∈ ∂E

∂ε +∂D

∂ε˙ and 0∈ ∂E

∂εe

+ ∂D

∂ε˙e

, (4)

whereσis the total Cauchy stress tensor. Using definition (2) of E andD, we get:

σ∈∂ϕ( ˙ε) +∂ϕp( ˙ε−ε˙e) and 0∈2µεe−∂ϕp( ˙ε−ε˙e). (5) The combination of the two previous relations leads to σ−2µεe ∈ ∂ϕ( ˙ε). Then, by using equation (11) from appendix A, and by introducing the pressure field p, we get the following expression of the total Cauchy stress tensor: σ=−p.I+ 2ηε˙+ 2µεewhen tr( ˙ε) = 0. Then, the second relation in (5) is equivalent

to ˙ε−ε˙e∈∂ϕ^∗_p(2µεe) where ϕ^∗_p is the dual of ϕp. Let us introduce the elastic stress tensor τ = 2µεe. Equation (10) in appendix A gives the expression for∂ϕ^∗, which yields:

1

2µτ˙+ max 0, |τd| −τ0

2k|τd|^r−1

!¹n

τ = ˙ε (6)

whereτd=τ−N¹ tr(τ)Idenotes the deviatoric part ofτ.

2.3 The system of equations

Since the material is considered in large deformations, we choose to use the Eulerian mathematical frame- work, more suitable for fluid flow computations. We assume that ˙ε=D(v) = ∇v+∇v^T

/2 is the rate of deformation, while the material derivative ˙τof tensorτin the Eulerian framework is expressed by the Gordon- Schowalter derivative [8] : ²τ=^∂τ_∂t+v.∇τ+τ W(v)−W(v)τ−a(τ D(v) +D(v)τ) whereW(v) = ∇v− ∇v^T

/2 is the vorticity tensor. The material parametera∈[−1,1] is associated with the Gordon-Schowalter’s deriva- tive. Whena= 0 we obtain the Jaumann derivative of tensors, whilea= 1 anda=−1 are associated with the upper and the lower convected derivatives, respectively.

The elastoviscoplastic fluid is then described by a set of three equations associated with three unknowns (τ,v, p):

the differential equation (6) is completed with the conservation of momentum and mass:













 1 2µ

τ2+ max

0, |τd| −τ0

2k|τd|ⁿ n¹

τ − D(v) = 0, ρ

∂v

∂t +v.∇v

−div (−pI+ 2ηD(v) +τ) = f, divv = 0,

whereρdenotes the constant density andf a known external force, such as the gravity. These equations are completed by some suitable initial and boundary conditions in order to close the system. For instance the initial conditionsτ(t= 0) =τ0andv(t= 0) =v0and the boundary conditionv= 0 on the boundary∂Ω are convenient. The total Cauchy stress tensor can be written as:

σ=−pI+ 2ηD(v) +τ.

Note that the incompressibility condition has been enforced by the definition ofϕin (3): ˙εis deviatoric and, since ˙ε =D(v), we get the incompressibility condition divv = tr ˙ε = 0. Conversely, the definition ofϕp

in (3) enforce that ˙εpis deviatoric. While ˙εe= ˙ε−ε˙pand the elastic stress tensorτ= 2µε˙eare expected to be deviatoric, the replacement of ˙τ by the Gordon-Schowalter tensor derivativeτ²do not preserve this property.

Instead, the non-deviatoric component of the elastic stress, that acts as an extra pressure component, decays over the elastic timescale.

2.4 Dimensionless formulation

LetU,Lbe some characteristic velocity and length of the flow, respectively. Let us introduceηp=k(L/U)¹⁻ⁿ, that has the dimension of a viscosity, and η0 =η+ηp that denotes the total viscosity. Let λ=ηp/µthat has the dimension of a time. Let T =L/U and Σ = (η+ηp)U/Lbe some characteristic time and stress, respectively. We introduce the following classical dimensionless numbers:

W e=λU

L , Bi= τ0L

η0U and Re=ρU L η0

n= 0.05

n= 0.2

n= 0.6

n= 1

n= 1.5 n= 2 n= 3 Bi

2α

κ

(s)

s/Bi

4 3

2 1

0 1

1/2

0

Figure 2: The Herschel-Bulkley plasticity criteria functionκn for variousnvalues.

which are the Weissenberg, Bingham and Reynolds numbers, respectively. The Weissenberg number is the ratio of the material time scaleλby the experiment characteristic timescaleL/Uwhile the Bingham number is the ratio of the yield stressσ0to the characteristic viscous stress. We use also the retardation parameter α=ηp/η0∈]0,1]. The problem reduces to that of finding dimensionless fields, also denoted by (τ,v, p) such that:











W eτ² +κn(|τd|)τ−2αD(v) = 0, Re

∂v

∂t +v.∇v

−div (−pI+ 2(1−α)D(v) +τ) = f, divv = 0,

(7)

whereκn denotes the plasticity criteria function : κn(s) = max

0, s−Bi (2α)¹⁻ⁿ sⁿ

¹n

, ∀s≥0 (8)

and f denotes some known dimensionless vector field. These equations are completed by the initial and boundary conditions. Notice that when W e= 0 and α= 1 the model reduces to the viscoplastic Herschel- Bulkley model and when Bi = 0 and n = 1 it reduces to the usual viscoelastic Oldroyd model [20, 25].

When bothW e=Bi= 0 andn= 1 the fluid is Newtonian and the set of equations reduces to the classical Navier-Stokes equations. Conversely, when bothW e6= 0 and Bi6= 0 the fluid is elastoviscoplastic.

Fig. 2 plots a scaled version of theκn function for various values ofn. Observe theκn(s) behavior ats=Bi.

The function is continuous and its left derivative is zero while its right derivative is also zero when n <1, it is 1/Biwhen n= 1 and +∞whenn >1. As a consequence, the function is smooth, i.e. its derivative is continuous, ats=Bi if and only if 0< n <1. Observe also that, when 0< n <1, thenκn is unbounded.

Conversely, whenn≥1, thenκn is bounded in R⁺ by a value denoted byκ^∗_n that depend uponn,Bi and

α. More precisely, the derivative ofκn is:

κ^′n(s) = (s−Bi)¹⁻

n n

(2α)^1−nrs²

Bi+ 1−n

n

s

, ∀s > Bi

and then, forn≥1,κn reaches its maximumκ^∗_n ats^∗= n

n−1

Biand

κ^∗_n=







1 whenn= 1

2α Bi

n−1

n (n−1)ⁿ⁻¹ⁿ

n whenn >1

These basic properties ofκnare essential for studying the model with practical examples. This is the subject of the following section.

3 Examples

3.1 Uniaxial elongation

The fluid is at rest at t = 0 and a constant elongational rate ˙ε0 is applied: the Weissenberg number is W e = λε˙0 and the Bingham number Bi = τ0/(η0ε˙0). All quantities presented in this paragraph are dimensionless.

The flow is three-dimensional and the dimensionless velocity gradient is ∇v= diag(1,−1/2,−1/2). The problem reduces to findτ11, τ22 andτ33 such that







W edτ11

dt + (κn(|τd|)−2aW e)τ11 = 2α, W edτjj

dt + (κn(|τd|) +aW e)τjj = −α, j= 2,3

with the initial conditionτ(t= 0) = 0. Sinceτ33=τ22 we have: |τd|= (2/3)¹²|τ11−τ22|. Sinceτ(0) = 0 and τ(t) is continuous, there existst0>0 such that when t∈[0, t0] we have|τd| ≤Bi and thusκn= 0: this is the linear flow regime. The eigenvalues of the system are 2aand−a. Fort > t0, the caseκn(|τd|)>0 occurs.

Whenn ≥1, since κn is bounded by κ^∗n, whenaW e > κ^∗n/2 the elastic stress is unbounded. Fig. 3a that plots theelongational viscosityηe⁺ normalized by 3η0(see e.g. [3, p. 132]). The ratio 3η0is the elongational viscosity for a Newtonian fluid. Note that for our dimensionless system we haveη⁺_e/(3η0) = (τ11−τ22)/3.

Conversely, when 0< n <1, sinceκn is strictly increasing and unbounded inR⁺, the factorκn(|τd|)−2aW e remains positive for large|τd|. Then the system is amortized and the solution always remains bounded. This feature when 0 < n < 1 is an important improvement to the case n = 1 that was previously considered in [26].

This drawback forn≤1 is still true for the Oldroyd viscoelastic model, i.e. whenn= 1 andBi= 0. In the context of viscoelastic models, some alternate constitutive equations that extend the Oldroyd model have been proposed, such as the Phan-Thien and Tanner model [21, 25] and the FENE-P model [4].

The stationary problem reduces to findτ11 andτ22 such that

(κn(|τd|)−2aW e)τ11 = 2α (κn(|τd|) +aW e)τ22 = −α

W e= 0.75

W e= 0.5

W e= 0.25

η

(t) 3η

(a)

t

10² 10¹

10⁰ 10⁻¹

10⁴

10²

10⁰

n= 1

n= 0.95

n= 0.9

n= 0.6 n= 0.2

η

(t) 3η

(b)

t

10² 10¹

10⁰ 10⁻¹

10⁴

10²

10⁰

Figure 3: Normalized elongational viscosityη_e⁺(t)/(3η0) for uniaxial elongation whenBi= 1,a= 1 andα= 1:

(a) influence ofW e forr= 2; (b) influence ofnforW e= 0.75.

Whena= 0 the stationary solution is independent ofW eandψ= (τ11−τ22)/2 satisfies: κn(|τd|)ψ= 3α/2.

Then ψ ≥ 0 and |τd|=p

8/3ψ. The previous equation leads to the following explicit expression of the steady elongational viscosityηe:

ηe

3η0

= 1−α+ (2/3)ψ= 1−α+ (2/3)¹⁻²ⁿα + (2/3)¹²Bi

Note that when a = 0, n = 1 and Bi = 0, we obtain the classical result ηe = 3η0 i.e. the elongational viscosity in the Newtonian case is three times the total viscosity.

Fig. 4 plots the normalized steady elongational viscosity whena= 1. The computation is no longer explicit fora6= 0, and requires numerical solution. We observe that it depends uponW e, while it was independent ofW e whena= 0. Moreover, whenn≥1, there exists a critical value ofW eupon which the elongational viscosity is no longer defined. Conversely, when 0< n <1, the elongational viscosity is defined for anyW e and is an increasing function ofW e.

3.2 Simple shear flow

The fluid is at rest att= 0 and a constant shear rate ˙γ0 is applied: the Weissenberg number is W e=λγ˙0

and the Bingham numberBi=τ0/(η0γ˙0).

The flow is two-dimensional and the dimensionless velocity gradient is constant: ∇v= ([0,1]; [0,0]). The problem reduces to findingτ11, τ22 andτ12, such that, for allt >0:

W ed dt



 τ11

τ22

τ12



+ (W eA^a+κn(|τd|).I)



 τ11

τ22

τ12



=



 0 0 α



,

n= 0.2 n= 0.5 n= 1

n= 1.5

η

3η

W e

10² 10¹

10⁰ 10⁻¹

10⁻² 10³

10²

10¹

10⁰

Figure 4: Normalized steady elongational viscosityηe/(3η0) fora= 1,Bi= 1, α= 1.

with the initial conditionτ(0) = 0, where

A^a =







0 0 −(1 +a)

0 0 1−a

1−a

2 −1 +a

2 0







with the initial conditionτ(0) = 0 and where|τd|²= (1/2) (τ11−τ22)²+ 2τ12². Letψ= (τ11−τ22)/2 be the dimensionless normal elastic stress difference. Thenτ11= 1 +a

2 ψandτ22=−1−a

2 ψ. The solution (τ12, ψ) is represented on Fig. 5. Sinceτ(0) = 0 andτ(t) is continuous, there existst0>0 such that when t∈[0, t0] we have|τd| ≤Biand thusκn(|τd|) = 0: this is the linear flow regime. The eigenvalues of the system are 0 and±i√

1−a². At t=t0, |τd|reachesBi. Then, fort > t0, the non-linear factor κn(|τd|)>0 occurs: the corresponding term reduces the growth of the solution, which now remains bounded for anyn >0.

Whena= 1 (see Fig. 5, where τ22 = 0) the solution tends to a constant value as t → ∞for any Bi≥0.

Observe on Fig. 5a thatτ12presents an overshoot that is more pronounced for small values ofn. Conversely, the steady value ofτ11−τ22 is monotonically increasing for any values ofn (Fig. 5b). The experimental evidence of non-vanishing first normal stress difference in shear flows can be found in [13] for a liquid foam in a Taylor-Couette geometry. Classical viscoplastic models such as the usual Bingham and Herschel-Bulkley models do not predict first normal stress difference in shear flows: this feature is due to viscoelasticity. Since the present model combines both viscoplasticity and viscoelasticity, it is able to predict yield stress behavior together with first normal stress difference in shear flows. In [5], the numerical prediction with the special case n = 1 of the present model has been compared for the Taylor-Couette geometry with experimental data from [13]. We show in Fig. 6 the steady shear viscosityηs/η0 (see e.g. [3, p. 104]). as a function of W e. Notice that the dimensionless steady shear viscosityηs/η0coincides with the dimensionless steady total shear stress σ12 = 1−α+τ12. The material presents a shear thinning character when 0 < n < 1 and a

n= 1 n= 0.5 n= 0.2 n= 0.05

τ

(a)

t

7 6 5 4 3 2 1 1.5

1

n= 1

n= 0.5 n= 0.2 n= 0.05

τ

− τ

2 (b)

t

7 6 5 4 3 2 1 3

2

1

Figure 5: Simple shear flow fora= 1,W e= 1,Bi= 1 andα= 1: (a)τ12; (b) (τ11−τ22)/2.

shear thickening character whenn >1. This feature is a second major improvement to the casen= 1 that was previously considered in [26]. For 0< n <1, the shear viscosity decreases monotonically when α= 1 (Fig. 6a), and tends to 1−αwhenα <1, due to the additional viscous contribution to the total shear stress σ12. The shear viscosity tends also to a plateau when n = 1 (Fig. 6b) and increases monotonically when n >1 (Fig. 6c). The value ofBicontrols the viscosity plateau at small values ofW ewhile it has less influence on the viscosity for large values ofW e. Note that this power-law behavior introduced here together with the Herschel-Bulkley viscoplastic model has been already introduced in the context of generalized Newtonian fluids by Carreau and Yasuda (see e.g. [4, p.171]) in order to be in agreement with shear rheology data.

Whena = 0 and Bi is small enough (see Fig. 7a, where τ22 =−τ11), the solution tends also to a steady value with decaying oscillations. These oscillations are due to presence of complex eigenvalues in the system.

Whena= 0 andBi becomes larger, oscillations no longer decay and instabilities appear, while the solution remains bounded (Fig. 7b). The Lissajou plot on Fig. 8 shows the asymptotical orbit of the solution for various values ofn. The orbit tangents the von Mises circle associated with|τd|=Bi(dotted lines). When the trajectory enters in the von Mises circle, the trajectory is exactly a circle, since the equations are linear whenκn(|τd|) = 0, while when its goes outside the circle, it decays, since the equations are non-linear. and κn(|τd|)>0. The asymptotical orbit is then completely included in the von Mises circle. We observe that the orbit in the Lissajou plot satisfies :

τ12=R0sin(ωt+ϕ) and ψ= τ11−τ22

2 = 1 +R0cos(ωt+ϕ)

where R0 is the radius of the orbit andϕ is the phase shift that depends upon the initial value. The von Mises criteria for the asymptotical orbit is maxt|τd(t)|=Bii.e.:

maxt (1 +R0cos(ωt))²+ (R0sin(ωt))²=Bi²/2 and thusR0=Bi/√

2−1. Finally, the asymptotical orbit is a circle of radiusR0and center (τ12, ψ) = (0,1) on the Lissajou plane. The power-law parameterndoes not change the orbit. It changes the speed of decay, as shown in Fig. 8. Finally, the critical Bingham number at which oscillations no longer decay isBic=√

2.

The shear viscosityη is no longer defined when a= 0 andW ebecomes large, since the solution for simple

Bi= 0 Bi= 1 Bi= 2 Bi= 4

η

η

(a)

W e

10² 10¹

10⁰ 10⁻¹ 10⁻² 10¹

10⁰ Bi= 0

Bi= 1 Bi= 2 Bi= 4

η

η

(b)

W e

10² 10¹

10⁰ 10⁻¹ 10⁻² 10¹

10⁰ Bi= 0

Bi= 1 Bi= 2 Bi= 4

η

η

(c)

W e

10² 10¹

10⁰ 10⁻¹ 10⁻² 10¹

10⁰

Figure 6: Steady shear viscosity fora= 1, α= 1 and (a)n= 0.5; (b) n= 1; (c)n= 1.5.

shear flow is no longer stationary and oscillates. The stability of the solution depends upon the values of the dimensionless parameters. This conditional stability is common in the context of viscoelastic models (Bi= 0): see e.g. [9] for Poiseuille flow and [27] for Couette-Taylor flows.

4 Conclusion

A new model for elastoviscoplastic fluid flows that is objective and satisfies the second law of thermodynamics is proposed in (7). It extends the previous elastoviscoplatic model introduced in [26] by introducing a power law indexn >0. As a major improvement, when 0< n <1 this model presents finite extensional properties and a shear thinning behavior. In [5], the numerical prediction with the present model and in the case when n= 1 has been compared for Taylor-Couette geometry with experimental data from [13]. Future work will extends quantitative comparisons between experimental measurements and numerical predictions to the case when 0< n <1. As many elastoviscoplatic materials have been found to exhibit shear thinning behavior, this model is a good candidate for numerical simulation of such materials in complex multi-dimensional geometries (see e.g. [6, 23, 24]).

A Technical details

A.1. Theϕp function – The subgradient∂ϕp, as introduced in (3), is defined for any tensorD by:

∂ϕp(D) = {τ, τ : (H −D) ≤ ϕp(H)−ϕp(D), ∀H}

= {τ, jp(D) ≤ jp(H), ∀H with tr(H) = 0 and tr(D) = 0},

with the notationjp(H) = _n^2k₊₁|H|ⁿ⁺¹+τ0|H|−τ:H. When the minimizerDofjpover the set{D,trD= 0} is non vanishing, it satisfies, from the theory of Lagrange multipliers:

∇jp(D)−p.I= 0 and tr(D) = 0,

τ

− τ

2

τ

(a)

t

20 10

0 2

1

0

τ

− τ

2

τ

(b)

t

40 30

20 10

0 2

1

0

-1

Figure 7: Simple shear flow fora= 0, n= 0.5,W e= 1 andα= 1: (a)Bi= 1; (b) Bi= 2.

where pis the Lagrange multiplier. Then 2k|D|ⁿ⁻¹D+τ0

D

|D|−τ−p.I= 0 and tr(D) = 0. Thus finally, the subgradient is:

∂ϕp(D) =











{τ, |τd| ≤τ0} whenD= 0,

τ, τ =−p.I+ 2k|D|ⁿ⁻¹D+τ0

D

|D|

whenD6= 0 and tr(D) = 0,

∅ otherwise,

(9)

whereτd denotes the deviatoric part ofτ. The dualϕ^∗_p ofϕp is then characterized by the Fenchel identity, that is, for anyτ∈∂ϕp(D), byϕ^∗_p(τ) =τ:D−ϕp(D). Moreover,τ∈∂ϕp(D) is equivalent to D∈∂ϕ^∗_p(τ).

Fromτ+p.I= (2k|D|ⁿ⁻¹+τ0/|D|)Dwe get|τd|= 2k|D|ⁿ+τ0and thus|D|= ((|τd| −τ0)/(2k))^1/n. Finally:

∂ϕ^∗_p(τ) = (

D, D= max

0, |τd| −τ0

2k|τd|^{n 1}/n

τd

)

, (10)

whereτd denotes the deviatoric part ofτ.

A.2. The ϕfunction – The function ϕ, as introduced in (3), is a particular case ofϕp withτ0 = 0, n= 1 andk=η. From (9), the subgradient is:

∂ϕ(D) =

{τ, τ=−p.I+ 2ηD} when tr(D) = 0,

∅ otherwise. (11)

References

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τ11−τ22

2

(a)

τ12

1 0

-1 2

1

0

τ11−τ22

2

(b)

τ12

1 0

-1 2

1

0

τ11−τ22

2

(c)

τ12

1 0

-1 2

1

0

Figure 8: Simple shear flow whena= 0,Bi= 2 andW e= 1 : flow instabilities and asymptotical orbit when (a)n= 0.5; (b)n= 1; (c)n= 2. The dotted line indicates the von Mises circle.

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